Direct and inverse asymptotic problems with high-frequency terms

Abstract

This paper is devoted to the first boundary value problem for the heat equation with a fast oscillating source. Direct and inverse problems are solved. The direct problem is to construct and justify an asymptotic expansion for the solution under appropriate assumptions. The inverse problem is to find the source if the value of two-term asymptotic expansion for solution at some point of space is given.

Keywords

asymptotics heat equation high-frequency inverse problem

1. Introduction

Construction and justification of asymptotic expansions for solutions of parabolic equations with oscillating in time coefficients is studied in a large number of works. In [5–8] the analysis is carried out for problems in bounded domains with Dirichlet boundary conditions, main attention is paid to time-periodic solutions. Initial boundary value problems are essentially more difficult and insufficiently studied (see, e.g., [7]). This paper is devoted to the first boundary value problem for the heat equation with a fast oscillating source $f (x) r (t, ω t)$ , $ω ≫ 1$ . Direct and inverse problems are solved. The direct problem is to construct and justify an asymptotic expansion for the solution $u_{ω} = u_{ω} (x, t)$ under appropriate assumptions. The inverse problem is to find the function $r (t, τ)$ if f (with $f (x_{0}) \neq 0$ ), and the value of two-term asymptotic expansion as $ω \to \infty$ for $u_{ω}$ at some point $x_{0} \in (0, π)$ , are given. Let us remark that the same problem with no highly oscillating parameter was studied in [1].

One of the interests of our paper is the reconstruction of a unique potential r from the two-term expansion of $u_{ω}$ . For the sake of simplicity, in this paper it is the one dimensional case that is studied, the generalization to $R^{n}$ being straightforward.

2. Direct problem. Two-term asymptotic approximation

Let Π the open rectangle defined as $\begin{matrix} {(x, t) : 0 < x < π; 0 < t < 1} \end{matrix}$ and denote by Γ its bottom and side parts of its boundary. Introduce also the notation $\begin{matrix} T = [0, 1] \times [0, + \infty) . \end{matrix}$ Consider the following parabolic initial-boundary value problem depending on the large parameter ω: $\begin{array}{rcl} (2.1) & \frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}} + f (x) r (t, ω t), (x, t) \in Π, \\ (2.2) & u |_{Γ} = 0 . \end{array}$ Let the functions f, r be continuous respectively, on $[0, π]$ and T. We assume that the function r is 2π-periodic with respect to $τ \in [0, + \infty)$ and represented in the form $\begin{matrix} r (t, τ) = r_{0} (t) + r_{1} (t, τ), \end{matrix}$ with $r_{0} \in C^{0} ([0, 1])$ , $r_{1} \in C^{2, 0} (T)$ , and $r_{1}$ with zero mean value with respect to τ, i.e., $\begin{matrix} {⟨ r_{1} (t, τ) ⟩}_{τ} \equiv \frac{1}{2 π} \int_{0}^{2 π} r_{1} (t, τ) d τ = 0 . \end{matrix}$ Suppose that $f \in C^{4 + α} ([0, π])$ , $α \in (0, 1)$ and is such that $\begin{matrix} (2.3) & f^{(2 n)} (0) = f^{(2 n)} (π) = 0, n = 0, 1, 2 . \end{matrix}$ Here $C^{l} ([0, π])$ , $C^{l, m} (T)$ , $C^{l, m} (\overline{Π})$ , with l, m nonnegative numbers, denote the standard Hölder spaces. Under the conditions formulated above, a two-term asymptotic approximation of the solution of (2.1)–(2.2) in $C^{2, 1} (\overline{Π})$ , with respect to a small parameter $ω^{- 1}$ , $ω ≫ 1$ is constructed and justified. Note that the requirements imposed on f and $r_{1}$ are connected with the asymptotic construction process in the space $C^{2, 1} (\overline{Π})$ .

The solution of (2.1)–(2.2) is sought in the form $\begin{matrix} (2.4) & u_{ω} (x, t) = U_{ω} (x, t) + W_{ω} (x, t), ω ≫ 1, \end{matrix}$ where $\begin{array}{rcl} (2.5) & U_{ω} (x, t) = u_{0} (x, t) + ω^{- 1} [u_{1} (x, t) + v_{1} (x, t, ω t)], ω ≫ 1, \\ (2.6) & u_{0} (x, t) = \sum_{n = 1}^{\infty} f_{n} sin n x \int_{0}^{t} e^{- n^{2} (t - s)} r_{0} (s) d s, f_{n} = \frac{1}{π} \int_{0}^{π} f (s) sin n s d s, \\ (2.7) & v_{1} (x, t, τ) = f (x) [\int_{0}^{τ} r_{1} (t, s) d s - {⟨ \int_{0}^{τ} r_{1} (t, s) d s ⟩}_{τ}], \\ (2.8) & u_{1} (x, t) = {⟨ \int_{0}^{τ} r_{1} (0, s) d s ⟩}_{τ} \cdot \sum_{n = 1}^{\infty} f_{n} sin n x e^{- n^{2} t} . \end{array}$ Expressions (2.6)–(2.8) will be obtained while proving the following theorem:

Theorem 1.
The solution $u_{ω} (x, t)$ of ( 2.1 )–( 2.2 ) is represented in the form ( 2.4 )–( 2.8 ), where $\begin{array}{rcl} (2.9) & {‖ W_{ω} (x, t) ‖}_{C (\overline{Π})} = O (ω^{- 2}), \\ (2.10) & {‖ W_{ω} (x, t) ‖}_{C^{2, 1} (\overline{Π})} = O (ω^{- 1}), ω \to \infty . \end{array}$

The proof will be given in Section 5.
3. Direct problem. N-term asymptotic approximation

The section is devoted to the construction of an N-term asymptotic expansion of the solution of (2.1)–(2.2) for any natural number N.

Let $N \in N$ , $N ⩾ 1$ , $f \in C^{2 N + α} ([0, π])$ , $α \in (0, 1)$ , $r_{0} \in C^{0} ([0, 1])$ , $r_{1} \in C^{N - 1, 0} (T)$ and 2π-periodic with respect to the second argument. Assume that $\begin{matrix} f^{(2 n)} (0) = f^{(2 n)} (π) = 0, n = \overline{0, N} . \end{matrix}$

We now introduce two types of problems.

The (A) problem is the following linear parabolic initial boundary value problem: $\begin{matrix} (3.1) & \{\begin{matrix} \frac{\partial}{\partial t} υ = \frac{\partial^{2}}{\partial x^{2}} υ + ψ (x, t), \\ υ |_{x = 0, π} = 0, υ |_{t = 0} = Ψ (x), \end{matrix} \end{matrix}$ with $ψ = ψ (x, t)$ defined on $x \in [0, π]$ , $t \in [0, 1]$ with $ψ \in C^{α, 0} (\overline{Π})$ , $Ψ = Ψ (x)$ defined on $x \in [0, π]$ , with $Ψ \in C^{2 + α} ([0, π])$ , $α \in (0, 1)$ . Moreover, it is assumed that $\begin{matrix} (3.2) & ψ (x, t) |_{x = 0, π} = Ψ (x) |_{x = 0, π} = Ψ^{″} (x) |_{x = 0, π} = 0 . \end{matrix}$

The (B) problem consists of finding $v = v (x, t, τ)$ , 2π-periodic with respect to τ and with zero mean value, solution of $\begin{matrix} (3.3) & \frac{\partial}{\partial τ} υ (x, t, τ) = μ (x, t, τ), \end{matrix}$ where $(x, t) \in [0, π] \times [0, 1]$ , $μ (x, t, \cdot) \in C^{0} ([0, \infty))$ and is 2π-periodic with respect to τ with zero mean value.

The solution of (2.1)–(2.2) is represented as $\begin{matrix} (3.4) & u_{ω} (x, t) = U_{ω}^{(N)} (x, t) + W_{ω}^{(N)} (x, t), ω ≫ 1, \end{matrix}$ with $\begin{matrix} (3.5) & U_{ω}^{(N)} (x, t) = u_{0} (x, t) + \sum_{k = 1}^{N - 1} ω^{- k} [u_{k} (x, t) + v_{k} (x, t, ω t)], \end{matrix}$ where the functions $v_{k}$ are 2π-periodic with zero mean value with respect to the third argument.

Theorem 2.
For any $N \in N$ , the representation ( 3.4 )–( 3.5 ) holds true, the term $U_{ω}^{(N)}$ can be constructed via the solution of a finite number of problems of type (A) and (B), and the summand $W_{ω}^{(N)}$ satisfies the estimates, $\begin{array}{rcl} {‖ W_{ω}^{(N)} (x, t) ‖}_{C (\overline{Π})} = O (ω^{- N}), \\ {‖ W_{ω}^{(N)} (x, t) ‖}_{C^{2, 1} (\overline{Π})} = O (ω^{- (N - 1)}), ω \to \infty . \end{array}$

The proof being analogous to the proof of Theorem 1 is omitted.
4. Inverse problem

Let us formulate an inverse problem to system (2.1)–(2.2). Consider the situation where f in (2.1)–(2.2) is known, while the function r is unknown and satisfies the conditions of Section 2. It is required to find r when the solution $u_{ω}$ at the point $x = x_{0}$ (with $f (x_{0}) \neq 0$ ) takes the form $\begin{matrix} (4.1) & u_{ω} (x_{0}, t, r) = ϕ_{0} (t) + ω^{- 1} ϕ_{1} (t, ω t) + ν_{ω} (t), \end{matrix}$ where $ϕ_{0} \in C^{1} ([0, 1])$ , $ϕ_{1} (t, τ) \in C^{1} (T)$ , $ν_{ω} \in C^{1} ([0, 1])$ and such that $\begin{array}{rcl} ϕ_{0} |_{t = 0} = ϕ_{1} |_{t = 0} = ν_{ω} |_{t = 0} = 0, \\ {‖ ν_{ω} (t) ‖}_{C [0, 1]} = O (ω^{- 2}), \\ {‖ \frac{d}{d t} ν_{ω} ‖}_{C [0, 1]} = O (ω^{- 1}), ω \to \infty . \end{array}$

Theorem 3.
The inverse problem is uniquely solvable.

The proof is given in Section 5. Remark.
As we will see from the proof of Theorem 3, the result holds if $\begin{matrix} \tilde{ϕ_{1}} (t, τ) = ϕ_{1} (t, τ) - {⟨ ϕ_{1} (t, τ) ⟩}_{τ}, \end{matrix}$ is known instead of $ϕ_{1}$ .

5. Main results proof

The next lemma is consequence of a well-known result.

Lemma 1.

The classical solution υ to system ( 3.1 ) is unique, belongs to $C^{2, 1} (\overline{Π})$ and satisfies the estimate $\begin{matrix} (5.1) & {‖ υ (x, t) ‖}_{C^{2, 1} (\overline{Π})} ⩽ c_{0} ({‖ ψ (x, t) ‖}_{C^{α, 0} (\overline{Π})} + {‖ Ψ (x) ‖}_{C^{2 + α} [0, π]}) . \end{matrix}$

Proof.

Without loss of generality, we consider $Ψ = 0$ . The decomposition of $ψ (\cdot, t)$ into Fourier series of sine functions $sin k x$ , $k \in N$ transforms problem (3.1) set in Π, into a Cauchy problem set in the region $(x, t) \in R \times [0, 1]$ with a homogeneous initial condition. The proof is completed by using the integral representation of the Cauchy problem solution via the Green’s function and applying known estimates (see [2], Chapter 1, Section 3). □

Proof of Theorem 1.

Substituting the expression $u_{ω}$ (2.4) into (2.1)–(2.2) gives $\begin{matrix} (5.2) & \{\begin{matrix} \frac{\partial U_{ω}}{\partial t} + \frac{\partial W_{ω}}{\partial t} = \frac{\partial^{2} U_{ω}}{\partial x^{2}} + \frac{\partial^{2} W_{ω}}{\partial x^{2}} + f (x) r (t, ω t), ω ≫ 1, \\ (U_{ω} + W_{ω}) |_{Γ} = 0 . \end{matrix} \end{matrix}$ The function $W_{ω} = W_{ω} (x, t)$ is represented in the form $\begin{matrix} (5.3) & W_{ω} (x, t) = ω^{- 2} [u_{2} (x, t) + v_{2} (x, t, ω t)] + W_{ω}^{(1)} (x, t), \end{matrix}$ with $v_{2} (x, t, τ)$ being a 2π-periodic with zero mean with respect to τ. Substituting (2.5) and (5.3) into (5.2), we obtain $\begin{matrix} (5.4) & \{\begin{matrix} \frac{\partial u_{0}}{\partial t} + \frac{\partial v_{1}}{\partial τ} + ω^{- 1} [\frac{\partial u_{1}}{\partial t} + \frac{\partial v_{1}}{\partial t} + \frac{\partial v_{2}}{\partial τ}] + ω^{- 2} [\frac{\partial u_{2}}{\partial t} + \frac{\partial v_{2}}{\partial t}] + \frac{\partial W_{ω}^{(1)}}{\partial t} \\ = \frac{\partial^{2} u_{0}}{\partial x^{2}} + ω^{- 1} [\frac{\partial^{2} u_{1}}{\partial x^{2}} + \frac{\partial^{2} v_{1}}{\partial x^{2}}] + ω^{- 2} [\frac{\partial^{2} u_{2}}{\partial x^{2}} + \frac{\partial^{2} v_{2}}{\partial x^{2}}] + \frac{\partial^{2} W_{ω}^{(1)}}{\partial x^{2}} + f (x) r (t, τ), \\ (u_{0} + ω^{- 1} (u_{1} + v_{1}) + ω^{- 2} (u_{2} + v_{2}) + W_{ω}^{(1)}) |_{Γ} = 0, τ = ω t, ω ≫ 1 . \end{matrix} \end{matrix}$ Equating the power-like terms of ω ( $k = 0, 1, 2$ ), and applying the averaging procedure with respect to ${⟨ \cdot ⟩}_{τ}$ $τ = ω t$ , we obtain the following problems: $\begin{array}{rcl} (5.5) & \{\begin{matrix} \frac{\partial u_{0}}{\partial t} = \frac{\partial^{2} u_{0}}{\partial x^{2}} + f (x) r_{0} (t), \\ u_{0} |_{Γ} = 0, \end{matrix} \\ (5.6) & \{\begin{matrix} \frac{\partial v_{1}}{\partial τ} = f (x) r_{1} (t, τ), \\ {⟨ v_{1} (x, t, τ) ⟩}_{τ} = 0, \end{matrix} \\ (5.7) & \{\begin{matrix} \frac{\partial u_{1}}{\partial t} = \frac{\partial^{2} u_{1}}{\partial x^{2}}, \\ u_{1} |_{t = 0} = - v_{1} |_{t = 0, τ = 0}, \\ u_{1} |_{x = 0} = u_{1} |_{x = π} = 0, \end{matrix} \\ (5.8) & \{\begin{matrix} \frac{\partial v_{2}}{\partial τ} = \frac{\partial^{2} v_{1}}{\partial x^{2}} - \frac{\partial v_{1}}{\partial t}, \\ {⟨ v_{2} (x, t, τ) ⟩}_{τ} = 0, \end{matrix} \\ (5.9) & \{\begin{matrix} \frac{\partial u_{2}}{\partial t} = \frac{\partial^{2} u_{2}}{\partial x^{2}}, \\ u_{2} |_{t = 0} = - v_{2} |_{t = 0, τ = 0}, \\ u_{2} |_{x = 0} = u_{2} |_{x = π} = 0 . \end{matrix} \end{array}$ Note that the obvious relations $v_{i} |_{x = 0, π} = \frac{\partial^{2}}{\partial x^{2}} v_{i} |_{x = 0, π} = 0$ ( $i = 1, 2$ ) are used in (5.7) and (5.9). It follows from (5.5)–(5.7) that the functions $u_{0}, v_{1}$ and $u_{1}$ have the form (2.6), (2.7), (2.8), respectively. Using relations (2.7), (5.8), we get $\begin{array}{rcl} v_{2} (x, t, τ) & = & f^{″} (x) [\int_{0}^{τ} ρ (t, s) d s - {⟨ \int_{0}^{τ} ρ (t, s) d s ⟩}_{τ}] \\ (5.10) & - f (x) [\int_{0}^{τ} \frac{\partial}{\partial t} ρ (t, s) d s - {⟨ \int_{0}^{τ} \frac{\partial}{\partial t} ρ (t, s) d s ⟩}_{τ}], \end{array}$ where $\begin{matrix} (5.11) & ρ (t, τ) = \int_{0}^{τ} r_{1} (t, s) d s - {⟨ \int_{0}^{τ} r_{1} (t, s) d s ⟩}_{τ} . \end{matrix}$ From (5.4)–(5.9) it follows that $\begin{matrix} (5.12) & \{\begin{matrix} \frac{\partial W_{ω}^{(1)}}{\partial t} = \frac{\partial^{2} W_{ω}^{(1)}}{\partial x^{2}} + ω^{- 2} [\frac{\partial^{2} v_{2}}{\partial x^{2}} - \frac{\partial v_{2}}{\partial t}], \\ W_{ω}^{(1)} |_{Γ} = 0 . \end{matrix} \end{matrix}$ Using relation (5.10), the maximum principle and Lemma 1, we obtain the estimates $\begin{array}{rcl} (5.13) & {‖ W_{ω}^{(1)} ‖}_{C (\overline{Π})} ⩽ c_{1} ω^{- 2}, \\ (5.14) & {‖ W_{ω}^{(1)} (x, t) ‖}_{C^{2, 1} (\overline{Π})} ⩽ c_{1} ω^{- 1}, \end{array}$ where $c_{1}$ does not depend on ω. Taking into account representations (5.3) and inequalities (5.13) and (5.14), we get the estimates from Theorem 1. This ends the proof of the theorem. □

Proof of Theorem 3.

It follows from Theorem 1 that for a function r satisfying the conditions of Section 2, problem (2.1)–(2.2) has a unique classical solution that is represented in form (2.4), i.e., $\begin{matrix} u_{ω} (x, t) = u_{0} (x, t) + ω^{- 1} [u_{1} (x, t) + v_{1} (x, t, ω t)] + W_{ω} (x, t), ω ≫ 1 . \end{matrix}$ Let function r be a solution of the inverse problem. By virtue of (4.1), $\begin{array}{rcl} u_{0} (x_{0}, t) + ω^{- 1} [u_{1} (x_{0}, t) + v_{1} (x_{0}, t, ω t)] + W_{ω} (x_{0}, t) \\ (5.15) & = ϕ_{0} (t) + ω^{- 1} ϕ_{1} (t, ω t) + ν_{ω} (t), ω ≫ 1 . \end{array}$ We now obtain an equivalent form of (5.15) by differentiating it with respect to t as follows: $\begin{array}{rcl} \frac{\partial}{\partial t} u_{0} (x_{0}, t) + \frac{\partial}{\partial τ} v_{1} (x_{0}, t, τ) |_{τ = ω t} + ω^{- 1} [\frac{\partial}{\partial t} u_{1} (x_{0}, t) + \frac{\partial}{\partial t} v_{1} (x_{0}, t, τ) |_{τ = ω t}] + \frac{\partial}{\partial t} W_{ω} (x_{0}, t) \\ (5.16) & = ϕ_{0}^{'} (t) + \frac{\partial}{\partial τ} ϕ_{1} (t, τ) |_{τ = ω t} + ω^{- 1} \frac{\partial}{\partial t} ϕ_{1} (t, τ) |_{τ = ω t} + \frac{d}{d t} ν_{ω} (t) . \end{array}$

Equating the power-like terms of $ω^{0}$ in Eq. (5.16) and averaging with respect to τ, we get $\begin{array}{rcl} (5.17) & \frac{\partial}{\partial t} u_{0} (x_{0}, t) = ϕ_{0}^{'} (t), \\ (5.18) & \frac{\partial}{\partial τ} v_{1} (x_{0}, t, τ) = \frac{\partial}{\partial τ} ϕ_{1} (t, τ) . \end{array}$ By virtue of Section 2, $\begin{matrix} \frac{\partial v_{1}}{\partial τ} = f (x) r_{1} (t, τ) . \end{matrix}$ Substituting it into (5.18) we obtain $\begin{matrix} (5.19) & f (x_{0}) r_{1} (t, τ) = \frac{\partial}{\partial τ} ϕ_{1} (t, τ) . \end{matrix}$ According to Section 2 the function $u_{0}$ is defined by (2.6). Differentiating the latter one with respect to t yields, $\begin{matrix} (5.20) & \frac{\partial}{\partial t} u_{0} (x_{0}, t) = f (x_{0}) r_{0} (t) + \int_{0}^{t} K (t - s) r_{0} (s) d s . \end{matrix}$ The function $K = K (t)$ defined as $\begin{matrix} K (t) = - \sum_{n = 1}^{\infty} n^{2} f_{n} e^{- n^{2} t} sin n x_{0}, \end{matrix}$ is continuous by virtue of the conditions imposed on f in Section 2.

Using (5.17) and (5.20) finally we have $\begin{matrix} (5.21) & f (x_{0}) r_{0} (t) + \int_{0}^{t} K (t - s) r_{0} (s) d s = ϕ_{0}^{'} (t) . \end{matrix}$

Equation (5.21) is a linear Volterra integral equation of the second kind, it has a unique continuous solution $r_{0}$ (cf. [9]). The unknown function $r_{1}$ is defined by formula (5.19) and according to the properties of $ϕ_{1}$ , belongs to $C^{1, 0} (T)$ . Therefore the function r is the only solution to the inverse problem. The statement of the theorem is proved. □

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Direct and inverse asymptotic problems with high-frequency terms

Abstract

Keywords

1. Introduction

2. Direct problem. Two-term asymptotic approximation

Theorem 3. The inverse problem is uniquely solvable. The proof is given in Section 5. Remark. As we will see from the proof of Theorem 3, the result holds if ϕ 1 ˜ ( t , τ ) = ϕ 1 ( t , τ ) − ⟨ ϕ 1 ( t , τ ) ⟩ τ , is known instead of ϕ 1 . 5. Main results proof

References